L(s) = 1 | + (−0.605 + 0.795i)2-s + (0.950 + 0.312i)3-s + (−0.266 − 0.963i)4-s + (0.981 − 0.189i)5-s + (−0.823 + 0.567i)6-s + (0.472 + 0.881i)7-s + (0.928 + 0.371i)8-s + (0.805 + 0.592i)9-s + (−0.444 + 0.895i)10-s + (−0.959 − 0.281i)11-s + (0.0475 − 0.998i)12-s + (−0.386 − 0.922i)13-s + (−0.987 − 0.158i)14-s + (0.991 + 0.126i)15-s + (−0.857 + 0.513i)16-s + (0.235 + 0.971i)17-s + ⋯ |
L(s) = 1 | + (−0.605 + 0.795i)2-s + (0.950 + 0.312i)3-s + (−0.266 − 0.963i)4-s + (0.981 − 0.189i)5-s + (−0.823 + 0.567i)6-s + (0.472 + 0.881i)7-s + (0.928 + 0.371i)8-s + (0.805 + 0.592i)9-s + (−0.444 + 0.895i)10-s + (−0.959 − 0.281i)11-s + (0.0475 − 0.998i)12-s + (−0.386 − 0.922i)13-s + (−0.987 − 0.158i)14-s + (0.991 + 0.126i)15-s + (−0.857 + 0.513i)16-s + (0.235 + 0.971i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.129812546 + 0.7619392062i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.129812546 + 0.7619392062i\) |
\(L(1)\) |
\(\approx\) |
\(1.094157519 + 0.5060927461i\) |
\(L(1)\) |
\(\approx\) |
\(1.094157519 + 0.5060927461i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 199 | \( 1 \) |
good | 2 | \( 1 + (-0.605 + 0.795i)T \) |
| 3 | \( 1 + (0.950 + 0.312i)T \) |
| 5 | \( 1 + (0.981 - 0.189i)T \) |
| 7 | \( 1 + (0.472 + 0.881i)T \) |
| 11 | \( 1 + (-0.959 - 0.281i)T \) |
| 13 | \( 1 + (-0.386 - 0.922i)T \) |
| 17 | \( 1 + (0.235 + 0.971i)T \) |
| 19 | \( 1 + (0.173 - 0.984i)T \) |
| 23 | \( 1 + (0.967 - 0.251i)T \) |
| 29 | \( 1 + (-0.975 - 0.220i)T \) |
| 31 | \( 1 + (-0.204 + 0.978i)T \) |
| 37 | \( 1 + (-0.939 - 0.342i)T \) |
| 41 | \( 1 + (-0.916 + 0.400i)T \) |
| 43 | \( 1 + (-0.939 + 0.342i)T \) |
| 47 | \( 1 + (0.527 - 0.849i)T \) |
| 53 | \( 1 + (0.991 - 0.126i)T \) |
| 59 | \( 1 + (0.580 - 0.814i)T \) |
| 61 | \( 1 + (-0.654 + 0.755i)T \) |
| 67 | \( 1 + (-0.327 + 0.945i)T \) |
| 71 | \( 1 + (-0.987 + 0.158i)T \) |
| 73 | \( 1 + (-0.701 - 0.712i)T \) |
| 79 | \( 1 + (0.873 + 0.486i)T \) |
| 83 | \( 1 + (0.0475 + 0.998i)T \) |
| 89 | \( 1 + (0.678 - 0.734i)T \) |
| 97 | \( 1 + (-0.745 - 0.666i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−26.62454911359342626300971219402, −26.01839884164575450914669186758, −25.19328867543058046968881949043, −24.11419437143210890016896783151, −22.81495110264985563309371844546, −21.50038183071544380356132684528, −20.70774914302781014376921370810, −20.39697899795228867897166262213, −18.891987593892179876247351771395, −18.45645651898912205819149928845, −17.37272090310831466294934394811, −16.47122574763579256592650979241, −14.78685178205880621932646111966, −13.72244312560244048481088289555, −13.2766279756002848075153634481, −11.98146602267889048631005860886, −10.61138499316471406980845007149, −9.81842806007798656117764056170, −8.990121169512878024848481051075, −7.67158711480318630883215546717, −7.04296865833950523526168124079, −4.92831810590061883240702828143, −3.53567389922116879520930295105, −2.3381286872163836831514664515, −1.45331369422162090211309478135,
1.70878090708323586927861324153, 2.837419808439254625077521221102, 4.992279116867438770437701555771, 5.55710534588359843052479331605, 7.12465366380328109119449897488, 8.35612040789090447677837282869, 8.86269504066364918614347821802, 9.992166954050652381988742212071, 10.74646429615032393541836114817, 12.86803623524651760918200192965, 13.62457849103589493106087025314, 14.89564771777284028412432661823, 15.226581560677851393989319740422, 16.44567837136304036366357045058, 17.59090262068419793606321032339, 18.35346053464989417486715604429, 19.27828976331344247894708086252, 20.41975150332165002080564486669, 21.34969198167326964194449351782, 22.178956005770476601639376177809, 23.78794425341448522195416988503, 24.708384092331354836453303837981, 25.202500048531807996359607259656, 26.08117610039468485716235741223, 26.79543752123387354403241852074